What about the strong clustering property and the OGP? First let us elucidate the relationship between the two
properties. The pair-wise (\(m=2\) ) OGP which holds for a single instance clearly implies the strong clustering
property: if every pair of \(\mu \) -optimal solutions is at distance at most \(\nu _1\)  or at least \(\nu _2>\nu _1\) ,
and the second case is non-vacuous, clearly the set of solutions is strongly clustered. It furthermore,
indicates that the diameter of each cluster is strictly smaller (at scale) than the distances between the clusters,
see Figures REF ,REF .
In fact this is precisely how the strong clustering property was discovered to begin
with in [1]} and [2]}. However,
the converse does not need to be the case,
as one could not rule out an example of a strong clustering property where the cluster
diameters are larger
than the distances between them, and, as a result,
the overlaps of pairs of solutions span a continuous intervals. At this
stage we are not aware of any such examples.
